Superconvergence Analysis of Finite Element Method for a Second-Type Variational Inequality

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Journal of Applied Mathematics Volume 2012, Article ID 156095,12pages doi:10.1155/2012/156095

Research Article

Superconvergence Analysis of Finite Element Method for a Second-Type Variational Inequality

Dongyang Shi,

¹

Hongbo Guan,

^{1, 2}

and Xiaofei Guan

³

1Department of Mathematics, Zhengzhou University, Zhengzhou 450001, China

2Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, China

3Department of Mathematics, Tongji University, Shanghai 200092, China

Correspondence should be addressed to Xiaofei Guan,[email protected] Received 10 May 2012; Accepted 14 October 2012

Academic Editor: Song Cen

Copyrightq2012 Dongyang Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper studies the finite elementFEapproximation to a second-type variational inequality.

The supe rclose and superconvergence results are obtained for conforming bilinear FE and nonconforming EQ^rotFE schemes under a reasonable regularity of the exact solutionu∈H^5/2Ω, which seem to be never discovered in the previous literature. The optimalL²-norm error estimate is also derived for EQ^rotFE. At last, some numerical results are provided to verify the theoretical analysis.

1. Introduction

Variational inequalityVItheory has been playing an important role in the obstacle problem, contact problem, elasticity problem, and so on 1. FE methods for solving VI problems have attracted more and more attentions. For example, as regards to the first type-VI case, the authors of 2used piecewise quadratic FE to approximate the obstacle problem and suggested the error order between the FE solution and the exact solution should beOh^3/2. The authors of3first obtained the error boundOh^3/2−ε for anyε >0for the above FE when the obstacle vanished. Then through a detailed analysis, the authors of4 obtained the same error bound as the ones of 3 under the hypothesis that the free boundary has finite length. Later, the authors of5obtained the same error bound as the ones of3for the same element without the hypothesis of finite length of the free boundary. Furthermore, 6investigated the Wilson’s element approximation to the obstacle problem and derived the error bound with orderOh. The authors of7obtained the same error estimate with order

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Ohon anisotropic meshes by making the full use of the bilinear part of the Wilson element, which relaxed the interpolation restriction and simplified the proofs of5,6. Recently, the authors of8proposed a class of nonconforming FE methods for the parabolic obstacle VI problem with moving grids and obtained the optimal error estimates on anisotropic meshes.

On the other hand, some studies9–11have been devoted to FE approximation to Signorini problem which arises in contact problems and obtained diﬀerent error estimates under diﬀerent assumptions. The authors of12derived the convergence result ofOh^3/4|logh|^1/4 if the displacement field is ofH²regularity and also showed that if stronger but reasonable regularity is available u ∈ W^2,p, p > 2, the above result can be improved to optimal order Oh. The authors of 13applied a class of Crouzeix-Raviart-type FEs to Signorini problem and obtainedOhorder estimate on anisotropic meshes. The authors of14used the bilinear FE to approximate the frictionless Signorini problem by virtue of the information on the contact zone and derived a superconvergence rate ofOh^3/2when the exact solution u∈ H^5/2Ω. The authors of15presented the nonconforming Carey FE approximation to the problem of14 and obtained the same convergence and superconvergence results are also obtained.

For the second type case, the authors of 16 proposed a Galerkin FE schemes for deriving a posteriori error estimates for a friction problem and a model flow of Bingham fluid. The authors of17considered the FE approximation to the plate contact problem and obtained some error estimates by employing the technique of mesh dependent norm.

In this paper, we will consider the following second type-VI problem18,19:

find u∈K^∗, such that au, v−u jv−ju≥

f, v−u

, ∀v∈K^∗, 1.1

whereΩ⊂R²is a bounded convex polygonal domain;K^∗is defined as follows:

K^∗

v∈H¹Ω|v 0,onΓ−Γd; v≥0, ∂v

∂n ≥0, v∂v

∂n 0, on Γd Γ⁰_d∪Γ_d

, 1.2

in whichΓ ∂Ω,Γd ⊂ΓandΓ⁰_d {x∈ Γd |vx 0},Γ_d {x ∈Γd | vx >0}.au, v

Ω∇u∇vμuvdx dy, μis a positive constant,f, v

Ωfv dx dy, jv

Γdψvds, and

ψt _t

0

ϕτdτ, ϕτ

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

g, τ ≥kg, τ

k, |τ| ≤kg,

−g τ ≤ −kg,

1.3

andg andkare positive constants.1.1may describe many practical engineering problems and attracts many scholars’ interests. For instance, the authors of20obtained theOh^1/2−ε error estimate of energy norm for linear FE; the authors of21got theOh^1/2error estimate in energy norm by improving the result of 20 for u ∈ H^3/2Ω; the authors of 22 derived the optimal Oh² error estimate of L² norm and Oh error estimate of energy norm whenu∈H²Ω. But all the above studies mentioned above only paid attention to the convergence analysis for the conforming FE with no consideration on the superconvergence

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property, although it is surely an interesting and useful phenomenon in scientific computing of industrial problems23.

In this paper, as a first attempt, we try to investigate the superconvergence of conforming and nonconforming FE schemes for problem1.1with a reasonable assumption ofu ∈ H^5/2Ω. The rest of this paper is organized as follows. In the next section, we give the equivalent form of1.1and the conforming bilinear FEsee14approximation of1.1.

Moreover, superclose result ofOh^3/2is derived under the broken energy norm. In Section3, the nonconforming EQ^rotFEsee26approximation is used, and the same superclose result is obtained under the energy norm; the optimal error estimate of L²-norm is also derived whenu∈H²Ω. In Section4, we construct a postprocessing interpolation operator to obtain the superconvergence properties. In Section5, we present some numerical results to verify the theoretical analysis.

2. The Equivalent Form and Conforming FE Scheme

It has been shown in21,22that1.1is equivalent to

findu∈K^∗, such that au, v

Γd

ϕuvds f, v

, ∀v∈K^∗, 2.1

and2.1has the unique solutionuinK^∗. It can be verified thatϕtsatisfies the following two properties: for alla, b∈R¹,

ϕa−ϕb≤ 1

k|a−b|, 2.2

ϕa−ϕb

a−b≥0. 2.3

LetT_h be a rectangular partition with a maximum sizehinx, yplane,K ∈ T_h a general element; V_h¹ and V_h² are the conforming bilinear FE space and the nonconforming EQ^rot FE space. We denote by Π¹_h and Π²_h the associated interpolation operators on V_h¹ and V_h², respectively. In the meantime, we denoteK_hⁱ by a convex set associated withK^∗inV_hⁱi 1,2 as follows:

K¹_h

v_h∈V_h¹ |v_h 0 on Γ−Γd

, K²_h

vh∈V_h²|

F

vhds 0, F⊂Γ−Γd,

F

vhds≥0, F ⊂Γd

,

2.4

whereF is an edge ofK. The following two lemmas will play an important role in the FE analysis, which can be found in14,24, respectively.

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Lemma 2.1. For allu∈H²Ω, F ⊂∂K, there holdsu−Πⁱ_hu0,F ≤Ch^3/2|u|2,K. Lemma 2.2. Letu∈H^5/2Ω, then forv_h∈K_h¹, there holds

∇

u−Π¹_hu , v_h

O h^3/2

|u|_5/2|vh|₁, 2.5

where|u|5/2

|α| 2

Ω|u^αϑ−u^αθ|²/|ϑ−θ|³dϑ dθ.

The corresponding conforming FE approximation version of2.1reads as find u∈K¹_h, such that

auh, v_h

Γd

ϕuhvhds f, v_h

, ∀vh∈K_h¹. 2.6

Theorem 2.3. Letu∈H^5/2Ωbe the exact solution of 1.1andu_h∈K_h¹the bilinear FE solution of 2.6, then there holds

Π¹_hu−uh

1≤ch^3/2|u|_5/2, 2.7

here and later,cis a generic positive constant, which is independent ofh,K, andu.

Proof. Subtracting2.1from2.6, then takingv v_hin it, one can get au−uh, vh

Γd

ϕu−ϕuh

vhds 0. 2.8

Letξ Π¹_hu−uhandη u−Π¹_hu. Takingvh ξin the above equation, there yields au−u_h, ξ

Γd

ϕu−ϕuh

ξds 0. 2.9

By the definition ofav, v, we have

|ξ|²₁≤aξ, ξ au−u_h, ξ−a η, ξ

−

Γd

ϕu−ϕuh

ξds−a η, ξ

−

Γd

ϕu−ϕ Π¹_hu

ξds−

Γd

ϕ

Π¹_hu

−ϕuh ξds−

∇η,∇ξ

−μ η, ξ

.

2.10

Noticing2.3, we have−

ΓdϕΠ¹_hu−ϕuhξds≤0; thus

|ξ|²₁≤I1I2, 2.11

in whichI1 −

Γdϕu−ϕΠ¹_huξds, I2 −∇η,∇ξ−μη, ξ.

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From2.2and Lemma2.1,I1can be estimated as

|I1| ≤ c k

Γd

η|ξ|ds≤η

0,Γdξ_0,Γ_d ≤ch^3/2|u|₂|ξ|₁. 2.12

Applying the interpolation theory and Lemma2.2, we get

|I2| ≤ch^3/2|u|_5/2|ξ|₁. 2.13 The desired result follows directly from the combination of2.12and2.13.

3. The Nonconforming FE Scheme

The corresponding nonconforming FE approximation scheme of2.1reads as find u∈K²_h, such that

a_huh, v_h

Γd

ϕuhvhds f, v_h

, ∀vh∈K²_h, 3.1

whereahu, v

K

K∇u∇vμuvdx dy.

First, we introduce the following Lemma 3.1, which can be found in25.

Lemma 3.1see25. Ifu∈H²Ω, vh∈K²_h, one has ∇

u−Π²_hu ,∇vh

0. 3.2

By using the similar technique in 26, one now states and proves the following important conclusion.

Lemma 3.2. For allu∈H^5/2Ω, vh∈K²_h, there holds

K

∂K

∂u

∂nv_hds≤ch^3/2|u|_5/2vh_h, 3.3 wherevhh

K∈Th|vh|²_1,K^1/2.

Proof. LetZ₁ x0−h_x, y₀−h_y, Z2 x0h_x, y₀−h_y, Z3 x0h_x, y₀h_y, andZ₄ x0−hx, y₀hybe the four vertices ofK,F_i Z_iZ_i1i 1,2,3,4, mod 4. We define operators P₀andP_0ias

P0v 1

|K|

K

v dx, P0iω 1

|Fi|

Fi

ω ds, 3.4

respectively, where|K|and|Fi|denote the measures ofKandF_i, respectively.

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It can be checked that

K

∂K

∂u

∂nv_hds

K

−

F1

∂u

∂yvh−P₀₁v_hdx

F2

∂u

∂xvh−P₀₂v_hdy

F3

∂u

∂yvh−P03vhdx−

F4

∂u

∂xvh−P04vhdy

F⊂Γd

F

∂u

∂nvhds

.

K

4 i 1

MiM.

3.5

By the definition ofP₀₁, we get

K

vh

x, y0−hy

−P01vh

x, y0−hy

dx dy

2hy

F1

vh

x, y0−hy

dx−4hxhy

|F1|

F1

vh

x, y0−hy

dx 0.

3.6

Noticing thatvh−P01vh|F1equalsvh−P03vh|F3and∂vh/∂xis only dependent onx, we can derive that

M₁M₃

_x₀_h_x

x0−hx

∂u

∂y

x, y₀h_y

−∂u

∂y

x, y₀−h_y

vh−P₀₁v_hdx _x₀_h_x

x0−hx

_y₀_h_y

y0−hy

∂²u

∂y² x, y

dy

vh−P01vhdx _x₀_h_x

x0−hx

_y₀_h_y

y0−hy

∂²u

∂y² −P0

∂²u

∂y²

vh−P01vhdy dx

∂²u

∂y² −P0∂²u

∂y² 0,K

vh−P01vh_0,K

≤ch^3/2|u|_5/2,K|vh|_1,K.

3.7

Similarly,M₂M₄ ≤ch^3/2|u|5/2,K|vh|1,K. By using the same technique as14,15,M can be estimated as

|M| ≤ch³|u|_5/2vh_h. 3.8

Thus the desired result follows.

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Theorem 3.3. Letu∈ H^5/2Ωbe the exact solution of 1.1anduh ∈K²_hthe nonconforming FE solution of 3.1. Then one has

Π²_hu−u_h

h≤Ch^3/2|u|_5/2. 3.9

Proof. Subtracting2.1from3.1gives ahu−uh, vh

Γd

ϕu−ϕuh

vhds

K

∂K

∂u

∂nvhds. 3.10 For convenience, we still denoteξ Π²_hu−uhandη u−Π²_hu. Takingvh Π²_hu−uh

in3.10yields

a_hu−u_h, ξ

Γd

ϕu−ϕuh

ξds

K

∂K

∂u

∂nξds. 3.11

By Lemma3.1, we can derive that ξ²_h≤ahξ, ξ ahu−uh, ξ−ah

η, ξ

−

Γd

ϕu−ϕuh

ξds−ah

η, ξ

K

∂K

∂u

∂nξds

−

Γd

ϕu−ϕ Π²_hu

ξds−

Γd

ϕ

Π²_hu

−ϕuh

ξds−μ η, ξ

K

∂K

∂u

∂nξds.

3.12 Noticing Lemma 3.2 and using the analysis technique of Theorem 2.3, one can immediately get the desired result.

Remark 3.4. As a by-product, if we assumeu∈H²Ωinstead ofu∈H^5/2Ω, the consistency error can be estimated as

K

∂K

∂u

∂nv_hds≤ch|u|₂vh_h, 3.13 which can be found in26. Then we can derive the following optimal error estimate:

u−uh_h≤Ch|u|₂. 3.14

Now we start to give theL²-norm estimate through a duality argument.

Theorem 3.5. Letu ∈ K²Ωanduh ∈ V_h²be the solutions of 1.1and3.1, respectively, there holds

u−uh₀≤Ch²|u|₂. 3.15

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Proof. Letω∈H²Ωbe the solution of the following auxiliary elliptic problem:

−wμw u−u_h, inΩ, w 0, onΓ−Γd,

∂w

∂n −βxw, onΓd,

3.16

in whichβx ϕu−ϕuh/u−u_h, then

w₂≤cu−u_h₀. 3.17

By3.16and Lemma3.1, we can derive that u−u_h²₀ u−u_h, u−u_h a_hu−u_h, w

Γd

βwu−u_hds

K

∂K

∂w

∂nu−u_hds ah

u−uh, w−Π²_hw ah

u−uh,Π²_hw

Γd

βwu−u_hds

K

∂K

∂w

∂nu−u_hds a_h

u−u_h, w−Π²_hw

−

Γd

ϕu−ϕuh

Π²_hw ds

Γd

βwu−u_hds

K

∂K

∂w

∂nu−uhds

K

∂K

∂u

∂nΠ²_hw ds ah

u−uh, w−Π²_hw 1

k

Γd

u−uh

w−Π²_hw ds

K

∂K

∂w

∂nu−uhds

K

∂K

∂u

∂n

w−Π²_hw ds

J1J2J3,

3.18

whereJ₁ a_hu−u_h, w−Π²_hw, J2 1/k

Γdu−u_hw−Π²_hwds,andJ₃

K

∂K∂w/∂nu− uhds

K

∂K∂u/∂nw−Π²_hwds. These three terms can be estimated one by one as follows.

By3.14,3.17, and the interpolation theory,J₁can be estimated as J₁

∇u−u_h,∇

w−Π²_hw μ

u−u_h, w−Π²_hw

≤ch²|u|₂|w|₂ch²u−uh₀|w|₂

≤ch²|u|₂u−uh₀ch²u−uh²₀.

3.19

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By the trace theorem,3.17, and Lemma2.1, one gets

J₂ ≤ 1

ku−u_h_0,Γ_du−Π²_hu

0,Γd

≤ch^5/2|u|₂|w|₂≤ch^5/2|u|₂u−u_h₀. 3.20

By3.13,3.14, and3.17, we have J₃≤ch|u|₂w−Π²_hw

hch|w|₂u−u_h_h≤ch²|u|₂|w|₂≤ch²|u|₂u−u_h₀. 3.21 The desired result follows the combination of the above estimates ofJ₁,J₂, andJ₃. Remark 3.6. As to theL²-norm error estimate of bilinear FE scheme, the readers may refer to 21,22.

4. The Global Superconvergence Result

In order to obtain the global superconvergence, we combine the four neighbouring elements K₁, K₂, K₃, K₄ ∈ T_h into one new rectangular elementK₀, whose four edges areL₁, L₂, L₃, andL₄.T_2h represents the corresponding new partition. For the conforming FE scheme, we construct the postprocessing operatorΠ¹_2hu|K0: CK0 → P2K0as follows:

Π¹_2hu Z_j

u Z_j

, j 1,2, . . . ,8, 4.1

in whichZ_jis the four vertices and four mid point of edges ofK₀. For the nonconforming FE scheme, we construct the postprocessingΠ²_2hoperator as

Π²_2hu|_K₀∈P₂K0, ∀K0∈T_2h,

Lj

Π²_2hu−u

ds 0, j 1,2,3,4,

K1∪K3

Π²_2hu−u dx 0,

K2∪K4

Π²_2hu−u

dx 0, ∀K0∈T2h.

4.2

It is easy to validate that the interpolation operator is well posed and has the following properties23:

Πⁱ_2hΠⁱ_hu Πⁱ_2hu, ∀u∈ H²Ω, Πⁱ_2hu−u

h≤ch^r|u|_r1, ∀u∈ H^r1Ω,0≤r ≤2, Πⁱ_2hv_h

h≤cvh_h, ∀vh∈Kⁱ_h.

4.3

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0

1 0

0.02

0.5 0.8

0.4 0.6 0.2 0.04

0.06 0.08 0.1

0 1

a

0 0.02 0.04 0.06 0.08 0.1

0.5 0.5

0

0 1

1 b

Figure 1: The conforming FE solutionaand the nonconforming FE solutionbon the 64×64 mesh.

Theorem 4.1. Ifu∈H^5/2Ωis the exact solution of 1.1,u_his the conforming or nonconforming FE solution. The following superconvergence result

u−Πⁱ_2hu_h

h≤ch^3/2|u|_5/2 4.4

holds.

Proof. By4.3, one gets Πⁱ_2hΠⁱ_hu−Πⁱ_2hu_h

h

Πⁱ_2hΠⁱ_hu−u_h

h≤cΠⁱ_hu−u_h

h≤ch^3/2|u|_5/2, Πⁱ_2hΠⁱ_hu−u

h

Πⁱ_2hu−u

h≤ch^3/2|u|_5/2.

4.5

NoticingΠⁱ_2huh−u Πⁱ_2huh−Πⁱ_2hΠⁱ_hu Πⁱ_2hΠⁱ_hu−u, the proof is completed.

5. Numerical Results

In this section, we will present an example to confirm the correctness of our theoretical analysis. In 1.1, we choose Ω 0,1× 0,1 with boundary ∂Ω Γ, μ 1, ϕu u,Γd {0} ×0,1, u|Γd x1 −1/2² −1/4, u|Γ−Γd 0. The right hand term f 1. Since there may be no exact solution to the above problem, we use the conforming FE solution on a suﬃcient refined mesh h 1/256 as the reference solution. Then we compare the conforming and nonconforming FE solutions see Figure 1 on the coarser meshes h 1/2,1/4,1/8,1/16,1/32,1/64with the reference one in Tables1and2.

From the above tables, we can see that the conforming and nonconforming FE solutions both converge. At the same time, the superconvergence results in our experiments are a little better than the theoretical ones. We may explain this phenomenon with some special properties of this nonconforming FE that we have not discovered.

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Table 1: The error estimates for conforming FE scheme.

h 1/2 1/4 1/8 1/16 1/32 1/64

Π¹_hu−uh

h 2.1780E−02 6.6596E−03 1.8851E−03 5.1760E−04 1.3861E−04 3.5405E−05

order / 1.8084 1.8796 1.9084 1.9324 1.9786

u−Π¹_2huh

h 5.3923E−03 1.5926E−03 4.1215E−04 1.0371E−04 2.5691E−05 6.1214E−06

order / 1.8401 1.9657 1.9935 2.0092 2.0486

Table 2: The error estimates for nonconforming FE scheme.

h 1/2 1/4 1/8 1/16 1/32 1/64

Π²_hu−uh

h 1.1264E−01 4.6572E−02 1.8679E−02 8.0004E−03 3.1977E−03 1.3906E−03

order / 1.5552 1.5790 1.5280 1.5817 1.5164

u−Π²_2huh

h 8.1523E−02 3.1059E−02 1.0260E−02 3.1725E−03 9.4359E−04 2.7340E−04

order / 1.6201 1.7399 1.7983 1.8336 1.8578

u−uh0 1.0995E−02 2.7109E−03 6.4263E−04 1.5941E−04 3.9258E−05 9.3430E−06

order / 2.0139 2.0539 2.0078 2.0151 2.0498

Acknowledgments

The first author was supported by the National Natural Science Foundation of China under Grant 10971203. The third author was supported by the National Natural Science Foundation of China under Grant 11126132. The authors would like to thank the referees for their valuable suggestions and corrections, which contribute significantly to the improvement of the paper.

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